 now next I will take you to an experiment, an experimental setup in a spallation neutron source now this is the schematic of prisma instrument at icis here we use time of flight technique so please note that we have a sample at a certain distance L maybe Ls if I may call it and then I have a bank of analyzers here at a certain distance with varying angle and then detectors so what it does actually you can see starting from here if this is the zero time at time t a neutron is detected now knowing this time t we know that from sample at this angle what is the energy or wavelength of the neutron that I am detecting at this detector because I know the analyzer angle I know what is the time so that means I can find out sample to detector I can calculate it because I know the wavelength or energy or velocity it is nothing but I can say L if I say Ls this I can say let us say L1 plus L2 sample to analyzer analyzer to detector I can let L1 plus L2 by V is the time taken for this part once I know the time taken for this part I can find out the time taken from the source to the sample by source I mean when I start the clock usually in a spallation neutron source the neutrons are produced by impinging protons on a target of high Z material not always uranium but high Z material which produces neutrons by spallation in a very short span of time with respect to the time scales that I am talking about that means moderation and the distances covered so then when the proton beam hits the spallation target I start my clock that is my zero time and when it is detected in this detector that is my T now knowing this T I can calculate out what is the T1 let us say for the given energy for this distance L1 plus L2 this is T1 then I can calculate out what is the time T0 from the source to the sample and then once I know T0 I can find out what is the wavelength of that neutron which has come here because in this case we do not have a monochromator look at this picture there is no monochromator so in case of a spallation neutron source what comes on the sample is a polychromatic beam it is a polychromatic beam so from the time of flight I find out what is the wavelength of the neutron that has reached the detector and just now I describe to you I know the outgoing energy so I can find out for the total path length what is the time taken which is T1 from the total time then I can find out what is the T0 and from that I can find out what is the lambda of the neutron now because it is a maxolian distribution the intensity of each lambda is not same so also there has to be a monitor detector here which finds out the intensity of the incoming neutrons and then the spectrum then we have to weigh each and every neutron with respect to this weightage because all the lambdas do not have same intensity falling on the sample we have to scale them as per their intensity so that means if I consider maximum intensity if I scale it to 1 then this one for example this is less number of neutrons this will be 1 divided by some fraction epsilon which will scale up this intensity because that one has low number of neutrons smaller number of neutrons compared to let us say this wavelength lambda lambda 1 has less number of neutrons compared to lambda 2 coming from another wavelength we have to scale them up or down to normalize the number of neutrons which are falling on the sample time of flight I can find out so I can scale the whole thing in terms of q and energy transfer h cross omega but this is the structure for a spallation neutron source for doing time of flight study of inelastic neutrons when we do diffraction work here then this analyzer will be absent and then again this part of scaling will remain same but we have we will have the total time and that I will convert it to lambda for the incoming neutron so this is a prisma spectrometer I will because I will show you a data which has been done in a reactor in a reactor like Dhruva then in a reactor like LLV in France and then it has also been done in ISIS at Rutherford-Appleton laboratory UK and the entire data ultimately stitched together to give the phonon dispersion relations I will come to that later after I start this discussion so the fact remains that phonon dispersion relation it's a different order of difficulty compared to if I talk about powder neutron diffraction or even single-tester neutron diffraction because in these experiments we need to be very careful about the incoming energy the outgoing energy and how to determine the h cross omega for a given momentum transfer for a given reciprocal lattice vector for a given phonon momentum vector ultimately this is the thing we want to find we want to find the dispersion relation as a function of q this is what our aim is but for that all this we have to find out so I have explained so now let me I have written it down earlier when we did diffraction experiment structure if you remember we wrote momentum transfer should be equal to a reciprocal lattice vector that's what our evolved plot was all about and also for a diffraction experiment Ki was equal to Kf magnitude was same because there is no energy transfer and evolved construction if I tell you that if this was my reciprocal lattice vector if you remember I start drawing a sphere of length k not ki but k of length k sphere which actually stops on a reciprocal lattice point here I can say 1 and then I draw a diameter I will draw a circle with this k as the radius and when I do that if I have this circle or in three dimensional sphere intercepting another reciprocal lattice point then you can see if this is Ki this is Kf then you can see Ki minus Kf it should be like this actually Ki minus Kf should be equal to the q reciprocal lattice vector and in this case because this circle has intercepted another reciprocal lattice point so that means q is equal to g which is the Bragg law in another format if I put Ki equal to 4 pi by lambda sin theta this is equal to 1 by 2d you will find I will get back to 4 pi by lambda sin theta equal to twice pi by d if I do then you can see that this is nothing but 2d sin theta equal to lambda so q equal to g is a more fundamental way of saying that this is satisfying Bragg's law and I will have a reflected beam in this direction if the incident beam is there this was eval construction now I have got a phonon associated with it so now my rules change a little bit so now I have earlier q was equal to g now q is equal to reciprocal lattice vector minus a phonon momentum vector because now this q is the signature of the phonon displacement q so let me now draw it once again for the case of phonons so let me see this is the reciprocal lattice vector let's say this is the reciprocal lattice vector let me consider that this is a q y and this is my origin and this is let us say this is my reciprocal lattice vector planes let's put it here these are my axes let us say x and y so my reciprocal lattice is oriented at some angle with respect to the chosen directions now this psi is basically the crystal that I am orienting with respect to the sample setup now this is if this is q which is the momentum transfer and then this is nothing but or let us say this is the k i which I am sorry this is k i the incident vector this is k f and then this is the vector q so this is k f the final energy k i and vectorially k i minus k f is equal to q k i minus k f is a momentum transfer q now this q this is a reciprocal lattice vector so this should be reciprocal lattice vector j g and minus a vector q so this is q large q this is a phoneme vector q this is the reciprocal lattice vector g and q plus q q plus q is the reciprocal lattice vector g this is the selection rule for inelastic neutrons scattering in a reciprocal lattice so now please know there is angle between k i and k f which is the angle between the incident direction and the outgoing direction which dictates q and this has to be associated with a phoneme vector q in some branch of the phoneme dispersion curve and q plus q should be equal to g a reciprocal vector reciprocal lattice vector in this diagram so that means i did not now if i consider this diagram this is a reciprocal lattice vector and this is not the first brillouis zone i can go over several brillouis zones depending on my q value and what g value i am choosing only this q q is less than equal to pi by a so the that means i can take it as a fraction of first brillouis zone so q can be maybe 0.1 0.2 of the reciprocal lattice vector now the reciprocal lattice vector is twice pi by a long so let me just define what i mean to say if this is my reciprocal lattice lattice i am just drawing a linear chain here linear chain here then here this is twice pi by a distance between the points q is somewhere less than pi by a so q when i say equal to 0.1 0.2 in reciprocal lattice vector this is a multiple of twice pi by a so q is a multiple of twice pi by either 0.1 or 0.2 but it has to be less than 0.5g because 0.5g means the boundary of the brillouis zone q should be less than that for all our calculations so i can in while i am choosing my q when i am choosing my q i can go several reciprocal lattice vectors then this is the phonon wave vector and q plus q should be equal to some reciprocal lattice vector g for the experiment so q plus q plus q equal to g so now i have shown it as drawing that this was k i this phi angle or i said psi in my previous drawing this is the angle of the reciprocal lattice with respect to the chosen system that means the sample axis the second axis can be rotated to give me this phi then this is my incident wave k i i mean the bold letters means their vectors so this is k i this is vector k f which is the final wave vector angle between them is theta this is chosen by choosing the angle of reflection and the analyzer position so it is chosen what angle i want to put the analyzer to detect the neutrons after that of course it will scatter further and go to the detector so this is the q vector and q and this is the small wave vector q together they should be equal to a reciprocal wave vector g now in this process for small phonon wave vectors phonon wave vector which is q i can measure at many q i can measure at g values and add them up so i have just shown a general relationship between i can measure at q1 momentum transfer for this q value with the reciprocal lattice vector g1 or i can measure even on a longer momentum transfer q2 for another reciprocal lattice vector g2 for phonon wave vector q so again and again i am telling this q i can use and change them to get various phonon wave vectors my dispersion relation is plotted against this q and these are the reciprocal wave vectors so let me just point out to you once again that you recall the eval construction that we had earlier when q was equal to g i also have a phonon momentum vector and then it is q plus q is equal to g now these are vector terms now for a linear lattice g is equal to twice pi by a and we can go several times the basis vector to do a measurement at a certain q then this is a general relationship now consider a longitudinal phonon in longitudinal phonon q and xi q is the momentum vector transfer and xi is the displacement of the atom for longitudinal phonon q and xi they will be parallel see this drawing looks like this so that means here it is on the same line q plus q and this is equal to g a momentum vector similarly for a transverse phonon the displacement is normal to the q or g values and it can be it should be looking like this the g and q the small q they are normal to each other because this is the reciprocal lattice vector is similar to what we have in real space the directions remain same and the transfer the wave vector the phonon wave vector or the displacement of the atoms is equal to the reciprocal wave vector and that is how this diagram shows that g and q they are perpendicular so in every experiment before into the experiment we have to choose the g values and the q values and then that will dictate what will be the q value in our measurement there is more to it and there are symmetry relations I will come to it in the next portion of my talk